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102       XIII   INTEGRATION

and Supp(/£)c Uar For this purpose we observe that if K is the support of
/, then there exist finitely many indices af e I (1 ^ i :g ri) such that the Uaf
cover K. Hence ((3.18.2) and (12.6.4)) there exist n continuous mappings
ht'.X-* [0, 1] such that Supp(Af) is compact and is contained in Uai for

n

1 <£ i ^ 72, and such that £ Af(jc) = 1 for all x e K. Then the functions/; =/A,

»=i
satisfy the required conditions.

This already proves the uniqueness of f* : for by definition we must have

To prove the existence of a linear form ft on ,3rc(X) whose restriction to
jTc(Ua) is /ia for each a e I, it is enough to establish the following assertion:
given two finite sequences (ffi)i^i^m and (hj)i£jgn of functions belonging to
JTC(X), such that Supp(#f) c U~ for l^i^m and Supp^) c U^^ for
1 ^j ^n, and such that

for all x e Supp(/), then we have

1-1             /=!

Now, we have

and therefore

m                          m    /   n                        \

E^«X/^-)= Z    T,H*t(f9ihj) •

i=J               i=l\J=l              /

Similarly,

But since Supp(/^f/i7.) is contained in UXi n Uftj, it follows from the hy-
potheses that ^ai(fyihj) = /*/»//<?!/*/), and our assertion follows.

It remains to be shown that the linear form \i so defined is a measure.
Let K be a compact subset of X, and define the Uai and the ht as at the be-
ginning of the proof. If Hf = Supp(Af), then by hypothesis there exists an write /in place of/| U when Supp(/) c U, and g in place of #u). The